numerical shooting methods for optimal boundary control and exact boundary … · 2015. 11. 16. ·...

INTERNATIONAL JOURNAL OF c⃝ 2016 Institute for ScientificNUMERICAL ANALYSIS AND MODELING Computing and InformationVolume 13, Number 1, Pages 122–144

NUMERICAL SHOOTING METHODS FOR OPTIMAL

BOUNDARY CONTROL AND EXACT BOUNDARY CONTROL

OF 1-D WAVE EQUATIONS

L. STEVEN HOU, JU MING, AND SUNG-DAE YANG∗

(Communicated by Max Gunzburger)

Abstract. Numerical solutions of optimal Dirichlet boundary control problems for linear andsemilinear wave equations are studied. The optimal control problem is reformulated as a system

of equations (an optimality system) that consists of an initial value problem for the underlying(linear or semilinear) wave equation and a terminal value problem for the adjoint wave equation.The discretized optimality system is solved by a shooting method. The convergence propertiesof the numerical shooting method in the context of exact controllability are illustrated through

computational experiments. In particular, in the case of the linear wave equation, convergentapproximations are obtained for both smooth minimum L2-norm Dirichlet control and generic,non-smooth minimum L2-norm Dirichlet controls.

Key words. Controllability, optimal control, wave equation, shooting method, finite difference

method.

1. Introduction

In this chapter we consider an optimal boundary control approach for solvingthe exact boundary control problem for one-dimensional linear or semilinear waveequations defined on a time interval (0, T ) and spatial interval (0, X). The exactboundary control problem we consider is to seek a boundary control g = (gL, gR) ∈L2(0,T) ⊂ [L2(0, T )]2 and a corresponding state u such that the following systemof equations hold:

(1)

utt − uxx + f(u) = V in Q ≡ (0, T )× (0, X) ,u|t=0 = u0 and ut|t=0 = u1 in (0, X) ,u|t=T =W and ut|t=T = Z in (0, X) ,u|x=0 = gL and u|x=1 = gR in (0, T ) ,

where u0 and u1 are given initial conditions defined on (0, X), W ∈ L2(0, X) andZ ∈ H−1(0, X) are prescribed terminal conditions, V is a given function defined on(0, T ) × (0, X), f is a given function defined on R, and g = (gL, gR) ∈ [L2(0, T )]2is the boundary control.

It is well known (see, e.g., [15, 16, 18, 19]) that when f = 0 (i.e., the equationis linear) and T is sufficiently large, the exact controllability problem (1) admitsat least one state-control solution pair (u,g); furthermore, the exact controller g

Received by the editors July 23, 2013, and accepted August 17, 2015.

2000 Mathematics Subject Classification. 93B40, 35L05, 65M06.∗ Corresponding author.The authors would like to thank the anonymous reviewers for constructive comments to improve

this paper. The corresponding author was supported by the Research for the Meteorological andEarthquake Observation Technology and Its Application of National Institute of MeteorologicalResearch. The second author was funded by National Natural Science Foundation of China undergrant number NSFC:91330104.

122

OPTIMAL AND EXACT BOUNDARY CONTROL OF 1-D WAVE EQUATIONS 123

having minimum boundary L2 norm is unique. Exact boundary controllability forsemilinear wave equations have also been established for certain asymptoticallylinear or superlinear f ; see, e.g., [4, 23, 24].

For the exact boundary controllability problem associated with the linear waveequation there are basically two classes of computational methods in the literature.The first class is HUM-based methods; see, e.g., [6, 9, 15, 17, 22]. The approximatesolutions obtained by the HUM-based methods in general do not seem to converge(even in a weak sense) to the exact solutions as the temporal and spatial grid sizestend to zero. Methods of regularization including Tychonoff regularization and fil-tering that result in convergent approximations were introduced in those papers onHUM-based methods. The second class of computational methods for boundarycontrollability of the linear wave equation was those based on the method proposedin [8]. One solves a discrete optimization problem that involves the minimization ofthe discrete boundary L2 norm subject to the undetermined linear system of equa-tions formed by the discretization of the wave equation and the initial and terminalconditions. This approach was implemented in [12]. The computational resultsdemonstrated the convergence of the discrete solutions when the exact minimumboundary L2 norm solution is smooth. In the generic case of a non-smooth exactminimum boundary L2 norm solution the computational results of [12] exhibitedat least a weak L2 convergence of the discrete solutions.

Although there are well-known theoretical results concerning boundary control-lability of semilinear wave equations (see, e.g., [4, 23, 24]), little seems to exist inthe literature about computational methods for such problems.

In this chapter we attempt to solve the exact controllability problems by anoptimal control approach. Precisely, we consider the following optimal controlproblem: minimize the cost functional

J0(u,g) =σ

2

∫ 10

|u(T, x)−W (x)|2 dx+ τ2

∫ 10

|ut(T, x)− Z(x)|2 dx

+1

2

∫ 10

(|gL|2 + |gR|2) dt(2)

subject to

(3)

utt − uxx + f(u) = V in Q ≡ (0, T )× (0, 1)u|t=0 = u0 and ut|t=0 = u1 in (0, 1)u|x=0 = gL and u|x=1 = gR in (0, T ) .

The optimal control problem is converted into an optimality system of equationsand this optimality system of equations will be solved by a shooting method.

The optimal control approach of this chapter provides an alternative method tothe two classes of methods mentioned in the foregoing for solving the exact control-lability problem for the linear wave equations; it also offers a systematic procedurefor solving exact controllability problems for the semilinear wave equations. Thecomputational solutions of this chapter obtained by an optimal control approachexhibit behaviors similar to those of the solutions obtained in [12]. Note that anoptimal solution exists even when the equation is not exactly controllable. Notealso that the solution methods in the literature for optimal control of PDEs can beutilized. and that there are certain intrinsic parallelisms to the algorithms studiedin this chapter.

124 L.S. HOU, J. MING, AND S.D. YANG

The shooting algorithms for solving the optimal control problem will be describedfor the slightly more general functional

J (u,g) =α2

∫ T0

∫ 10

|u− U |2 dx dt+ σ2

∫ 10

|u(T, x)−W (x)|2 dx

+τ

2

∫ 10

|ut(T, x)− Z(x)|2 dx+1

2

∫ 10

(|gL|2 + |gR|2) dt(4)

where the term involving (u − U) reflects our desire to match the candidate stateu with a given U in the entire domain Q. Our computational experiments of theproposed numerical methods will be performed exclusively for the case of α = 0.

The rest of this chapter is organized as follows. In Section 2 we establish theequivalence between the limit of optimal solutions and the minimum boundary L2

norm exact controller; this justifies the use of the optimal control approach forsolving the exact control problem. In Section 3 we formally derive the optimalitysystem of equations for the optimal control problem and discuss the shooting algo-rithm for solving the optimality system. In Section 4 We state the discrete versionof the shooting algorithm for solving the discrete optimality system. Finally in Sec-tions 5 and 6 we present computations of certain concrete controllability problemsby the shooting method for solving optimal control problems.

2. The solution of the exact controllability problem as the limit of opti-mal control solutions

In this section we establish the equivalence between the limit of optimal solu-tions and the minimum boundary L2 norm exact controller. We will show that ifα = 0, σ → ∞ and τ → ∞, then the corresponding optimal solution (ûστ , ĝστ ) con-verges weakly to the minimum boundary L2 norm solution of the exact boundarycontrollability problem (1). The same is also true in the discrete case.

Theorem 2.1. Assume that the exact boundary controllability problem (1) admitsa unique minimum boundary L2 norm solution (uex,gex). Assume that for every(α, σ, τ) ∈ {0} × R+ × R+ (where R+ is the set of all positive real numbers,) thereexists a solution (uστ ,gστ ) to the optimal control problem (17). Then

(5) ∥gστ∥L2(Σ) ≤ ∥gex∥L2(Σ) ∀ (α, σ, τ) ∈ {0} × R+ × R+ .Assume, in addition, that for a sequence {(σn, τn)} satisfying σn → ∞ and τn → ∞,(6) uσnτn ⇀ u in L

2(Q) and f(uσnτn)⇀ f(u) in L2(0, T ; [H2(Ω) ∩H10 (Ω)]∗) .

Then

(7) gσnτn ⇀ gex in [L2(0, T )]2 and uσnτn ⇀ uex in L

2(Q) as n→ ∞ .Furthermore, if (6) holds for every sequence {(σn, τn)} satisfying σn → ∞ andτn → ∞, then(8) gστ ⇀ gex in [L

2(0, T )]2 and uστ ⇀ uex in L2(Q) as σ, τ → ∞ .

Proof. Since (uστ ,gστ ) is an optimal solution, we have that

σ

2∥uστ (T )−W∥2L2(0,1) +

τ

2∥∂tuστ (T )− Z∥2H−1(0,1) +

1

2∥gστ∥2L2(Σ)

= J (uστ ,gστ ) ≤ J (uex,gex) =1

2∥gex∥2L2(Σ)

so that (5) holds,

(9) uστ |t=T →W in L2(0, X) and (∂tuστ )|t=T → Z in H−1(0, X) as σ, τ → ∞ .


Let {(σn, τn)} be the sequence in (6). Estimate (5) implies that a subsequence of{(σn, τn)}, denoted by the same, satisfies

(10) gσnτn ⇀ g in [L2(0, T )]2 and ∥g∥L2(0,T ) ≤ ∥gex∥L2(0,T ) .

(uστ ,gστ ) satisfies the initial value problem in the weak form:∫ T0

∫ X0

uστ (vtt − vxx) dx dt+∫ T0

∫ X0

[f(uστ )− V ]v dx dt

+

∫ T0

gστ |x=X(∂xv)|x=X dt−∫ T0

gστ |x=0(∂xv)|x=0 dt

+

∫ X0

(v∂tuστ )|t=T dx−∫ X0

v|t=0u1 dx−∫ X0

(uστ∂tv)|t=T dx

+

∫ X0

(u0∂tv)|t=0 dx = 0 ∀ v ∈ C2([0, T ];H2 ∩H10 (0, X))

(11)

where gστ |x=0 denotes the first component of gστ and gστ |x=X the second compo-nent of gστ . Passing to the limit in (11) as σ, τ → ∞ and using relations (9) and(10) we obtain:∫ T

0

∫ X0

u(vtt − vxx) dx dt+∫ T0

∫ X0

[f(u)− V ]v dx dt+∫ T0

gR(∂xv)|x=X dt

−∫ T0

gL(∂xv)|x=0 dt+∫ X0

v|t=TZ(x) dx−∫ X0

v|t=0u1 dx−∫ X0

W (x)(∂tv)|t=T dx

+

∫ X0

(u0∂tv)|t=0 dx = 0 ∀ v ∈ C2([0, T ];H2 ∩H10 (0, X)) .

The last relation and (10) imply that (u,g) is a minimum boundary L2 normsolution to the exact control problem (1). Hence, u = uex and g = gex so that (7)and (8) follows from (6) and (10). �

Remark 2.2. If the wave equation is linear, i.e., f = 0, then assumption (6) isredundant and (8) is guaranteed to hold. Indeed, (11) implies the boundedness of{∥uστ∥L2(Q)} which in turn yields (6). The uniqueness of a solution for the linearwave equation implies (6) holds for an arbitrary sequence {(σn, τn)}.

Theorem 2.3. Assume that

: i) for every (α, σ, τ) ∈ {0} × R+ × R+ there exists a solution (uστ ,gστ ) tothe optimal control problem (17);

: ii) the limit terminal conditions hold:(12)uστ |t=T →W in L2(0, X) and (∂tuστ )|t=T → Z in H−1(0, X) as σ, τ → ∞ ;

: iii) the optimal solution (uστ ,gστ ) satisfies the weak limit conditions asσ, τ → ∞:

(13) gστ ⇀ g in L2(0, T ) , uστ ⇀ u in L

2(Q) ,

and

(14) f(uστ )⇀ f(u) in L2(0, T ; [H2(Ω) ∩H10 (Ω)]∗)

for some g ∈ L2(0, T ) and u ∈ L2(Q).


Then (u,g) is a solution to the exact boundary controllability problem (1) with gsatisfying the minimum boundary L2 norm property. Furthermore, if the solutionto (1) admits a unique solution (uex,gex), then

(15) gστ ⇀ gex in [L2(0, T )]2 and uστ ⇀ uex in L

2(Q) as σ, τ → ∞ .

Proof. (uστ ,gστ ) satisfies (11). Passing to the limit in that equation as σ, τ → ∞and using relations (12), (13) and (14) we obtain:∫ T

0

∫ X0

u(vtt − vxx) dx dt+∫ T0

∫ X0

[f(u)− V ]v dx dt+∫ T0

gR(∂xv)|x=X dt

−∫ T0

gL(∂xv)|x=0 dt+∫ X0

v|t=TZ(x) dx−∫ X0

v|t=0u1 dx−∫ X0

W (x)(∂tv)|t=T dx

+

∫ X0

(u0∂tv)|t=0 dx = 0 ∀ v ∈ C2([0, T ];H2 ∩H10 (0, X)) .

This implies that (u,g) is a solution to the exact boundary controllability problem(1).

To prove that g satisfies the minimum boundary L2 norm property, we proceedsas follows. Let (uex,g) denotes a exact minimum boundary L

2 norm solution tothe controllability problem (1). Since (uστ ,gστ ) is an optimal solution, we havethat

σ

2∥uστ −W∥2L2(0,X) +

τ

2∥∂tuστ − Z∥2H−1(0,X) +

1

2∥gστ∥2L2(0,T )

= J (uστ ,gστ ) ≤ J (uex,gex) =1

2∥gex∥2L2(0,T )

so that

∥gστ∥L2(0,T ) ≤ ∥gex∥L2(0,T ) .Passing to the limit in the last estimate we obtain

(16) ∥g∥L2(0,T ) ≤ ∥gex∥L2(0,T ) .

Hence we conclude that (u,g) is a minimum boundary L2 norm solution to theexact boundary controllability problem (1).

Furthermore, if the exact controllability problem (1) admits a unique minimumboundary L2 norm solution (uex,gex), then (u,g) = (uex,gex) and (15) follows fromassumption (13). �

Remark 2.4. If the wave equation is linear, i.e., f = 0, then assumptions i) and(14) are redundant.

Remark 2.5. Assumptions ii) and iii) hold if gστ and uστ converges pointwise asσ, τ → ∞.

Remark 2.6. A practical implication of Theorem 2.3 is that one can prove theexact controllability for semilinear wave equations by examining the behavior of asequence of optimal solutions (recall that exact controllability was proved only forsome special classes of semilinear wave equations.) If we have found a sequenceof optimal control solutions {(uσnτn ,gσnτn)} where σn, τn → ∞ and this sequenceappears to satisfy the convergence assumptions ii) and iii), then we can confidentlyconclude that the underlying semilinear wave equation is exactly controllable andthe optimal solution (uσnτn ,gσnτn) when n is large provides a good approximationto the minimum boundary L2 norm exact controller (uex,gex).


3. An optimality system of equations and a continuous shooting method

Under suitable assumptions on f and through the use of Lagrange multiplierrules, the optimal control problem

(17) minimize (4) with respect to the control g subject to (3)

may be converted into the following system of equations from which an optimalsolution may be determined:

utt − uxx + f(u) = V in (0, T )× (0, X) ,u|x=0 = gL , u|x=1 = gR , u(0, x) = u0(x), ut(0, x) = u1(x) ,ξtt − ξxx + f ′(u)ξ = −α(u− U) in (0, T )× (0, X) ,ξ|x=0 = 0, ξ|x=1 = 0 ,ξ(T, x) = −τA−1(ut(T, x)− Z(x)) ξt(T, x) = −σ(u(T, x)−W (x)) ,gL = −ξx|x=0 , and gR = ξx|x=1 ,

where the elliptic operator A : H10 (0, X) → H−1(0, X) is defined by Av = vxx for allv ∈ H10 (0, X). By eliminating gL and gR in the system we arrive at the optimalitysystem

utt − uxx + f(u) = V in (0, T )× (0, X) ,u|x=0 = −ξx|x=0 , u|x=1 = ξx|x=1 ,u(0, x) = u0(x), ut(0, x) = u1(x) ,

ξtt − ξxx + f ′(u)ξ = −α(u− U) in (0, T )× (0, X) ,ξ|x=0 = 0, ξ|x=1 = 0 ,ξ(T, x) = −τA−1(ut(T, x)− Z(x)) , ξt(T, x) = −σ(u(T, x)−W (x)) .

(18)

Derivations and justifications of optimality systems are discussed in [13] for thelinear case and in [14] for the semilinear case.

The computational algorithm we propose in this chapter is a shooting method forsolving the optimality system of equations. The basic idea for a shooting methodis to convert the solution of a initial-terminal value problem into that of a purelyinitial value problem (IVP); see, e.g., [2] for a discussion of shooting methods forsystems of ordinary differential equations. The IVP corresponding to the optimalitysystem (18) is described by

utt − uxx + f(u) = V in (0, T )× (0, X) ,

u|∂Ω =∂ξ

∂ν

∣∣∣∂Ω, u(0, x) = u0(x), ut(0, x) = u1(x) ;

ξtt − ξxx + f ′(u)ξ = −α(u− U) in (0, T )× (0, X) ,ξ|∂Ω = 0, ξ(0, x) = ω(x), ξt(0, x) = θ(x) ,

(19)

with unknown initial values ω and θ. Then the goal is to choose ω and θ such thatthe solution (u, ξ) of the IVP (19) satisfies the terminal conditions

F1(ω, θ) ≡ ∂xxξ(T, x) + τ(ut(T, x)− Z(x)) = 0 ,F2(ω, θ) ≡ ξt(T, x) + σ(u(T, x)−W (x)) = 0 .

(20)

A shooting method for solving (18) can be described by the following iterations:

choose initial guesses ω and θ;for iter = 1, 2, · · · ,maxiter


solve for (u, ξ) from the IVP (19)update ω and θ.

A criterion for updating (ω, θ) can be derived from the terminal conditions (20).A method for solving the nonlinear system (20) (as a system for the unknownsω and θ) will yield an updating formula; for instance, the well-known Newton’smethod may be invoked.

choose initial guesses ω and θ;for iter = 1, 2, · · · ,maxiter

solve for (u, ξ) from the IVP (19)update ω and θ:(ωnew, θnew) = (ω, θ)− [F ′(ω, θ)]−1F (ω, θ);if F (ωnew, θnew) = 0, stop; otherwise, set (ω, θ) = (ωnew, θnew).

A discussion of Newton’s method for an infinite dimensional nonlinear systemcan be found in many functional analysis textbooks, and for the suitable assumptionconvergence of Newton iteration for the optimality system is guaranteed.

4. The discrete shooting method

The shooting method described in Section (3) must be implemented discretely.We discretize the spatial interval [0, 1] into 0 = x0 < x1 < x2 < · · · < xI < xI+1 = 1with a uniform spacing h = 1/(I + 1) and we divide the time horizon [0, T ] into0 = t1 < t2 < t3 < · · · < tN = T with a uniform time stepping δ = T/(N − 1). Weuse the explicit, central difference scheme to approximate the initial value problem(19):

u1i = (u0)i, u2i = (u0)i + δ(u1)i , ξ

1i = ωi, ξ

2i = ξ

1i + δθi , i = 1, 2, · · · , I ;

un+1i = −un−1i + λu

ni−1 + 2(1− λ)uni + λuni+1

− δ2f(uni ) + δ2V (tn, xi), i = 1, 2, · · · , I ,ξn+1i = −ξ

n−1i + λξ

ni−1 + 2(1− λ)ξni + λξni+1

− δ2f ′(uni )ξni + δ2α(uni − U(tn, xi)), i = 1, 2, · · · , I

un+10 = −ξn+11 − ξ

n+10

h, un+1I+1 = −

ξn+1I+1 − ξn+1I

h.

(21)

where λ = (δ/h)2 (we also use the convention that ξn0 = ξnI+1 = 0.) The gist of a

discrete shooting method is to regard the discrete terminal conditions

F2i−1 ≡ξNi − ξ

N−1i

δ+ σ(uNi −Wi) = 0 ,

F2i ≡ξNi−1 − 2ξNi + ξNi+1

h2+ τ(

uNi − uN−1i

δ− Zi) = 0 , i = 1, 2, · · · , I

(22)

as a system of equations for the unknown initial condition ω1, θ1, ω2, θ2, · · · , ωm, θm.Similar to the continuous case, the discrete shooting method consists of the

following iterations:

choose discrete initial guesses {ωi}Ii=1 and {θ}Ii=1;for iter = 1, 2, · · · ,maxiter

solve for {(uni , ξni )}i=I,n=Ni=1,n=1 from the discrete IVP (21)

update {ωi}Ii=1 and {θi}Ii=1.The initial conditions {ωi}Ii=1 and {θi}Ii=1 are updated by Newton’s method

applied to the discrete nonlinear system (22). This requires the calculations of


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−13

−12

−11

−10

−9

−8

−7

−6

−5

−4

−3

x0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−10

−5

0

5

10

15

20

25

30

35

40

x

Figure 1. left - u0 , right - u1 given in (24). h = 1/256.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−5

−4

−3

−2

−1

0

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

00.20.40.6

0.811.21.4

1.61.8

−15

−10

−5

0

5

10

x

t

Figure 2. left - exact control , right - exact u. with initial data(24). h = 1/32.

Table 1. Results of computational experiments for the minimumL2(Σ)-norm case for the examples with initial data (24).

h 1/16 1/32 1/64 1/128 1/256 1/512 1/1024

∥gh∥L2(Σ) λ = 1 5.9339 6.0294 6.0825 6.1103 6.1244 6.1316 6.1352

∥gh∥L2(Σ) λ = 7/8 5.9682 6.0468 6.0917 6.1454 6.1262 6.1325 6.1356∥gh−g∥L2(Σ)

∥g∥L2(Σ)λ = 1 6.93% 3.35% 1.63% 0.79% 0.37% 0.18% 0.09%

∥gh−g∥L2(Σ)∥g∥L2(Σ)

λ = 7/8 7.53% 4.26% 2.88% 10.15% 0.35% 0.17% 0.08%

00.2

0.40.6

0.81

0

0.5

1

1.5

2−15

−10

−5

0

5

10

x

uh

t 00.2

0.40.6

0.81

0

0.5

1

1.5

2−15

−10

−5

0

5

10

x

u

t

Figure 3. left - approximate control uh , right - exact u withinitial data (24). h = 1/256. λ = 1.

partial derivatives. By denoting

qnij = qnij(ω1, θ1, ω2, θ2, · · · , ωI , θI) =

∂uni∂ωj

, rnij = rnij(ω1, θ1, ω2, θ2, · · · , ωI , θI) =

∂uni∂θj

,

ρnij = ρnij(ω1, θ1, ω2, θ2, · · · , ωI , θI) =

∂ξni∂ωj

, τnij = τnij(ω1, θ1, ω2, θ2, · · · , ωI , θI) =

∂ξni∂θj

,


0 1 1.8−5

−4

−3

−2

−1

0

1

2

3

4

5

t0 0.5 1

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

x0 0.5 1

−12

−10

−8

−6

−4

−2

0

2

x

0 1 1.8−5

−4

−3

−2

−1

0

1

2

3

4

5

x0 0.5 1

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

x0 0.5 1

−12

−10

−8

−6

−4

−2

0

2

x

0 1 1.8−5

−4

−3

−2

−1

0

1

2

3

4

5

x0 0.5 1

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

x0 0.5 1

−12

−10

−8

−6

−4

−2

0

2

x

0 1 1.8−5

−4

−3

−2

−1

0

1

2

3

4

5

t0 0.5 1

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

x0 0.5 1

−12

−10

−8

−6

−4

−2

0

2

x

Figure 4. left - approximate control gh and g, middle - approx-imate uh and target W , right - approximate uht and target Z withinitial data (24). h = 1/16, 1/32, 1/64, 1/1024 from top to bottomrespectively. λ = 1.

we obtain the following Newton’s iteration formula:

(ωnew1 , θnew1 , ω

new2 , θ

new2 , · · · , ωnewI , θnewI )T

=(ω1, θ1, ω2, θ2, · · · , ωI , θI)T

− [F ′(ω1, θ1, ω2, θ2, · · · , ωI , θI)]−1F (ω1, θ1, ω2, θ2, · · · , ωI , θI)

where the vector F and Jacobian matrix J = F ′ are defined by

F2i−1 =ξNi − ξ

N−1i

δ+σ(uNi −Wi) , F2i =

ξNi−1 − 2ξNi + ξNi+1h2

+τ(uNi − u

N−1i

δ−Zi) ,

J2i−1,2j−1 =ρNij − ρ

N−1ij

δ+ σqNij , J2i−1,2j =

τNij − τN−1ij

δ+ σrNij ,

J2i,2j−1 =ρN(i−1)j − 2ρ

Nij + ρ

N(i+1)j

h2+τ

δ(qNij − qN−1ij ) ,

J2i,2j =τN(i−1)j − 2τ

Nij + τ

N(i+1)j

h2+τ

δ(rNij − rN−1ij ) .


Moreover, by differentiating (21) with respect to ωj and θj we obtain the equationsfor determining qij , rij , ρij and τij :

q1ij = 0, q2ij = 0 , r

1ij = 0 , r

2ij = 0 ,

ρ1ij = δij , ρ2ij = δij , τ

1ij = 0 , τ

2ij = δδij ,

i, j = 1, 2, · · · , I ;

qn+1ij = −qn−1ij + λq

ni−1,j + 2(1− λ)qnij + λqni+1,j

− δ2f ′(uni )qnij , i, j = 1, 2, · · · , I ,rn+1ij = −r

n−1ij + λr

ni−1,j + 2(1− λ)rnij + λrni+1,j

− δ2f ′(uni )rnij , i, j = 1, 2, · · · , I ,ρn+1ij = −ρ

n−1ij + λρ

ni−1,j + 2(1− λ)ρnij + λρni+1,j

+ δ2αqNij − δ2[f ′(uni )]ρni − δ2[f ′′(uni )qNi ]ξni , i, j = 1, 2, · · · , I ;τn+1ij = −τ

n−1ij + λτ

ni−1,j + 2(1− λ)τnij + λτni+1,j

+ δ2αrNij − δ2[f ′(uni )]ρni − δ2[f ′′(uni )rNi ]ξni , i, j = 1, 2, · · · , I ;

(23)

where δij is the Chronecker delta. Thus, we have the following Newton’s-method-based shooting algorithm:

Algorithm – Newton method based shooting algorithm with central finite differ-ence approximations of the optimality system

choose initial guesses ωi and θi, i = 1, 2, · · · , I;% set initial conditions for u and ξfor i = 0, 2, · · · , I + 1

u1i = (u0)i, u2i = (u0)i + δ(u1)i,

for i = 1, 2, · · · , Iξ1i = ωi, ξ

2i = ξ

1i + δθi;

% set initial conditions for qij , rij , ρij , τij

for j = 1, 2, · · · , Ifor i = 1, 2, · · · , I

q1ij = 0, q2ij = 0, r

1ij = 0, r

2ij = 0,

ρ1ij = 0, ρ2ij = 0, τ

1ij = 0, τ

2ij = 0;

ρ1jj = 1, ρ2jj = 1, τ

2jj = δ,

% Newton iterations

for m = 1, 2, · · · ,M% solve for (u, ξ)for n = 2, 3, · · · , N − 1

un+1i = −un−1i + λu

ni−1 + 2(1− λ)uni + λuni+1 − δ2f(uni ) +

δ2V (tn, xi),ξn+1i = −ξ

n−1i + λξ

ni−1 + 2(1− λ)ξni + λξni+1 − δ2f ′(uni )ξni

+δ2α(uni − U(tn, xi));% solve for q, r, ρ, τfor j = 1, 2, · · · , I

for n = 2, 3, · · · , N − 1for i = 2, · · · , N − 1

qn+1ij = −qn−1ij + λq

ni−1,j + 2(1− λ)qnij + λqni+1,j

−δ2f ′(uni )qnij ,rn+1ij = −r

n−1ij + λr

ni−1,j + 2(1− λ)rnij + λrni+1,j

−δ2f ′(uni )rnij ,


ρn+1ij = −ρn−1ij + λρ

ni−1,j + 2(1− λ)ρnij + λρni+1,j

+δ2αqnij−δ2[f ′(uni )]ρnij−δ2[f ′′(uni )qnij ]ξni ,τn+1ij = −τ

n−1ij + λτ

ni−1,j + 2(1− λ)τnij + λτni+1,j

+δ2αrnij−δ2[f ′(uni )]ρnij−δ2[f ′′(uni )rnij ]ξni ;% (we need to build into the algorithm the fol-

lowing:

qn+10j = −ρn+11jh , r

n0j = −

τn+11jh , r

nI+1,j = −

τn+1Ijh

qnI+1,j = −ρn+1Ijh , ρ

n0j = τ

n0j = ρ

nI+1,j =

τnI+1,j = 0.)

% evaluate F and F ′

for i = 1, 2, · · · , IF2i−1 =

ξNi −ξN−1i

δ + σ(uNi −Wi),

F2i =ξNi−1−2ξ

Ni +ξ

Ni+1

h2 + τ(uNi −u

N−1i

δ − Zi);for j = 1, 2, · · · , I

J2i−1,2j−1 =ρNij−ρ

N−1ij

δ +σqNij , J2i−1,2j =

τNij −τN−1ij

δ +

σrNij

J2i,2j−1 =ρN(i−1)j−2ρ

Nij+ρ

N(i+1)j

h2 +τδ (q

Nij − q

N−1ij ),

J2i,2j =τN(i−1)j−2τ

Nij +τ

N(i+1)j

h2 +τδ (r

Nij − r

N−1ij );

solve Jc = −F by Gaussian eliminations;for i = 1, 2, · · · , I

ωnewi = ωi + c2i−1, θnewi = θi + c2i;

if maxi |ωnewi − ωi|+maxi |θnewi − θi| < tol, stop;otherwise, reset ωi = ω

newi and θi = θ

newi , i = 1, 2, · · · , I;As in the continuous case, we have the following convergence result for the shoot-

ing algorithm which follows from standard convergence results for Newton’s methodapplied to finite dimensional systems of nonlinear equations.

Remark 4.1. The algorithms we propose are well suited for implementations on aparallel computing platform such as a massive cluster of processors. The shootingalgorithms of this chapter can be regarded as a generalization of their counterpartfor systems of ODE (see, e.g., [2].) There has been a substantial literature on theparallelization of shooting methods for ODEs [3, 10, 11]; these results will be helpfulin parallelizing the shooting algorithms of this chapter.

Table 2. Results of computational experiments for the minimumL2(Σ)-norm case for Examples I with initial data (26) and forλ = 1, 4/5.

h 1/32 1/64 1/128 1/256 1/512 1/1024

∥gh∥L2(Σ) λ = 1 0.12934 0.12908 0.12906 0.12907 0.12908 0.12909∥gh∥L2(Σ) λ = 4/5 0.15941 0.15269 0.14522 0.14216 0.13907 0.13622

5. Computational experiments for controllability of the linear wave equa-tion

We will apply Algorithm 1 to the special case of f = 0, V = 0, W = 0, Z = 0,α = 0 and σ, τ >> 1. In other words, we will approximate the null controllability


0 1 1.8−5

−4

−3

−2

−1

0

1

2

3

4

5

t0 0.5 1

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x0 0.5 1

−2

−1

0

1

2

3

4

5

6

x

0 1 1.8−5

−4

−3

−2

−1

0

1

2

3

4

5

t0 0.5 1

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x0 0.5 1

−2

−1

0

1

2

3

4

5

6

x

0 1 1.8−5

−4

−3

−2

−1

0

1

2

3

4

5

t0 0.5 1

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x0 0.5 1

−2

0

5

10

15

20

25

30

35

x

0 1 1.8−5

−4

−3

−2

−1

0

1

2

3

4

5

t0 0.5 1

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

x0 0.5 1

−2

−1

0

1

2

3

4

5

6

x

Figure 5. left - approximate control gh and g, middle - approx-imate uh and target W , right - approximate uht and target Zwith initial data (24). h = 1/16, 1/32, 1/64, 1/1024 from top tobottom respectively. λ = 7/8.

Table 3. Results of computational experiments for the minimumL2(Σ)-norm case for Examples II with initial data (26) and forλ = 1, 7/8.

h 1/16 1/32 1/64 1/128 1/256 1/512 1/1024

∥gh∥L2(Σ) λ = 1 0.6838 0.6388 0.6162 0.6049 0.5992 0.5963 0.5949∥gh∥L2(Σ) λ = 7/8 0.6734 0.6348 0.6138 0.6039 0.5988 0.5963 0.5949

Table 4. Results of computational experiments for the minimumL2(Σ)-norm case for for Examples III with initial data (26) andfor λ = 1, 7/8.

h 1/16 1/32 1/64 1/128 1/256 1/512 1/1024

∥gh∥L2(Σ) λ = 1 1.4277 1.3187 1.2605 1.2303 1.2149 1.2071 1.2032∥gh∥L2(Σ) λ = 7/8 1.3932 1.3007 1.2493 1.2252 1.2124 1.2065 1.2028


0 0.5 1−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

t0 0.5 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

x0 0.5 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

x

0 0.5 1−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

t0 0.5 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

x0 0.5 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

x

0 0.5 1−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

t0 0.5 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

x0 0.5 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

x

0 0.5 1−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

t0 0.5 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

x0 0.5 1

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

x

Figure 6. left - approximate control gh , middle - approximateuh and target W , right - approximate uht and target Z with ini-tial data (26-I). h = 1/16, 1/32, 1/64, 1/1024 from top to bottomrespectively. λ = 1.

Table 5. Results of computational experiments for the minimumL2(Σ)-norm case for Examples I, II, III in (27) with initial data Iin (26) and for λ = 1.

h 1/16 1/32 1/64 1/128

I ∥gh∥L2(Σ) 0.08084810765 0.08038960736 0.08021218880 0.08013073451count 17 16 16 17

II ∥gh∥L2(Σ) 0.07346047350 0.07314351955 0.07307515741 0.07306230119count 8 8 10 12

III ∥gh∥L2(Σ) 0.07438729446 0.07404916115 0.07397863882 0.07396393744count 6 5 5 5

problem for the linear wave equation by optimal control problems. We will testour algorithm with a smooth example (i.e., the continuous minimum boundaryL2 norm controller g and the corresponding state u are smooth) and with threegeneric examples. It was reported in [12] that the discrete minimum boundary L2

norm controllers converge strongly to the continuous minimum boundary L2 norm


0 0.5 1−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t0 0.5 1

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x0 0.5 1

−60

−50

−40

−30

−20

−10

0

x

0 0.5 1−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t0 0.5 1

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x0 0.5 1

−60

−50

−40

−30

−20

−10

0

x

0 0.5 1−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t0 0.5 1

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x0 0.5 1

−60

−50

−40

−30

−20

−10

0

x

0 0.5 1−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

t0 0.5 1

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x0 0.5 1

−2000

−1500

−1000

−500

0

500

x

Figure 7. left - approximate control gh , middle - approximateuh and target W , right - approximate uht and target Z with ini-tial data (26-I). h = 1/16, 1/32, 1/64, 1/1024 from top to bottomrespectively. λ = 4/5.

Table 6. Results of computational experiments for the minimumL2(Σ)-norm case for Examples I, II, III in (27) with initial data IIin (26) and for λ = 1.

h 1/16 1/32 1/64 1/128

I ∥gh∥L2(Σ) 0.45499490909 0.43856890841 0.43072624972 0.42688986194count 12 15 16 14

II ∥gh∥L2(Σ) 0.45794379129 0.43786262977 0.42823496745 0.42350319979count 12 12 12 17

III ∥gh∥L2(Σ) 0.45251184147 0.43223256110 0.42248163895 0.41769256787count 6 6 6 6

controller for the smooth example and converge weakly in the generic case. Thediscrete optimal solutions found by Algorithm 1 will exhibit similar behaviors.

5.1. An example with known smooth exact solution. A smooth exact solu-tion to the minimum boundary L2-norm controllability problem was constructed in[12] by using Fourier series in a way similar to that used in [6]. Suppose that Q =


0 1 1.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t0 0.5 1

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

x0 0.5 1

−12

−10

−8

−6

−4

−2

0

2

x

0 1 1.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t0 0.5 1

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

x0 0.5 1

−12

−10

−8

−6

−4

−2

0

2

x

0 1 1.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t0 0.5 1

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

x0 0.5 1

−25

−20

−15

−10

−5

0

5

x

0 1 1.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t0 0.5 1

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

x0 0.5 1

−400

−350

−300

−250

−200

−150

−100

−50

0

50

x

Figure 8. left - approximate control gh , middle - approximateuh and target W , right - approximate uht and target Z with ini-tial data (26-II). h = 1/16, 1/32, 1/64, 1/1024 from top to bottomrespectively. λ = 1.

Table 7. Results of computational experiments for the minimumL2(Σ)-norm case for Examples I, II, III in (27) with initial dataIII in (26) and for λ = 1.

h 1/16 1/32 1/64 1/128

I ∥gh∥L2(Σ) 0.94846408635 0.86623499117 0.82305989083 0.80084943961count 11 11 11 11

II ∥gh∥L2(Σ) 0.99946692390 0.90706619363 0.85837779589 0.83325287647count 13 16 18 20

III ∥gh∥L2(Σ) 0.95205362894 0.86225101645 0.81393537311 0.78872084130count 6 6 6 6

(0, 7/4)× (0, 1) and Σ = (0, 7/4)×{0, 1}. Let ψ0(t, x) = −√2π cosπ(t− 14 ) cos 2πx


0 1 1.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t0 0.5 1

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

x0 0.5 1

−30

−25

−20

−15

−10

−5

0

5

x

0 1 1.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t0 0.5 1

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

x0 0.5 1

−60

−50

−40

−30

−20

−10

0

10

x

0 1 1.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t0 0.5 1

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

x0 0.5 1

−120

−100

−80

−60

−40

−20

0

20

x

0 1 1.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t0 0.5 1

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

x0 0.5 1

−300

−250

−200

−150

−100

−50

0

50

x

0 1 1.8−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t0 0.5 1

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

x0 0.5 1

−600

−500

−400

−300

−200

−100

0

100

x

Figure 9. left - approximate control gh , middle - approximateuh and target W , right - approximate uht and target Z with ini-tial data (26-II). h = 1/64, 1/128, 1/256, 1/512, 1/1024 from top tobottom respectively. λ = 7/8.

and

ψ1(t, x) =

[2√2π(T − t) sinπ(t− 1

4)− 10

3

√2 sinπ(t− T )

]sinπx

+∑

p≥3 and p odd

4√2

p2 − 1

[3p

p2 − 4cosπ(t− 1

4) + sin pπ(t− T )

]sin pπx .

Then, set the initial conditions

(24) u0(x) = ψ0(0, x) + ψ1(0, x) and u1(x) =∂ψ0∂t

(0, x) +∂ψ1∂t

(0, x) .


0 1 1.8−1

−0.5

0

0.5

1

1.5

2

t0 1 1.8

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t0 0.5 1

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x0 0.5 1

−40

−35

−30

−25

−20

−15

−10

−5

0

5

x

0 1 1.8−1

−0.5

0

0.5

1

1.5

2

t0 1 1.8

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t0 0.5 1

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x0 0.5 1

−40

−35

−30

−25

−20

−15

−10

−5

0

5

x

0 1 1.8−1

−0.5

0

0.5

1

1.5

2

t0 1 1.8

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t0 0.5 1

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x0 0.5 1

−40

−35

−30

−25

−20

−15

−10

−5

0

5

x

0 1 1.8−1

−0.5

0

0.5

1

1.5

2

t0 1 1.8

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

t0 0.5 1

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

x0 0.5 1

−600

−500

−400

−300

−200

−100

0

100

x

Figure 10. gL, gR, uh(x, T ), and uht (x, T ) from left to right with

initial data (26-III). h = 1/16, 1/32, 1/64, 1/1024 from top to bot-tom respectively. λ = 1.

The computation of u0 and u1 involve the summation of infinite trigonometricseries. Figure 1 and Figure 2 provides plots of u0 and u1, and the exact control andexact solution respectively. Note that initial conditions vanish at the boundary, anddue to symmetry, we have gL(t) = u(t, 0) = u(t, 1) = gR(t). i.e the controls at twosides of Q are the same. It is worth noting that u0 is a Lipschitz continuous functionbut does not belong to C1[0, 1] and u1 is a bounded function but does not belong toC0[0, 1]. For the initial data (24), it can be shown that u(t, x) = ψ0(t, x) +ψ1(t, x)is the exact solution having minimum boundary L2-norm of the controllabilityproblem given by the first three equations in (1) provided f = 0, V = 0. Let g bethe corresponding exact Dirichlet control given by restricting u(t, x) to the lateralsides Σ. i.e g(t) = (gL(t), gR(t)) = (u(t, 0), u(t, 1)), and

(25) gL(t) = gR(t) = −√2π cosπ

(t− 1

4

).

For future reference, note that ∥g∥L2(Σ) =√2π2( 74 +

12π ) ≈ 6.13882.

We apply our numerical method to this example. Computational experimentswere carried out for h = 1/16, 1/32, 1/64, 1/128, 1/256, 1/512, and 1/1024 withλ = 1 and λ = 7/8 respectively, so that the stability condition is satisfied.


0 1 1.8−1.5

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x


initial data (26-III). h = 1/64, 1/128, 1/256, 1/512, 1/1024 fromtop to bottom respectively. λ = 7/8.

The results of our computational experiments are summarized in Table 1, wheregh are the computed approximations of the exact solutions g. All norms werecalculated by linearly interpolating the nodal values of gh. From this table, it seemsthat gh converges to g in the L2(Σ)-norm at a rate of roughly 1. In order to visualizethe convergence of our method as h becomes smaller, in Figure 3 we provide plotsof the exact solution u and the corresponding computed discrete solutions uh forh = 1/256 with λ = 1. Figure 4 and 5 are plots of the exact solution g and thecorresponding computed discrete solutions gh, a given functionW and approximatesolutions uh, and a given function Z and approximate solutions uht for h = 1/16,h = 1/32, h = 1/64, and h = 1/1024.

It seems that our method produces (pointwise) convergent approximations forboth λ = 1 and λ = 7/8 without the need for regularization. This should be


0 0.5 1 1.5 2 2.5 3−0.5

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f(u) = sinu and initial data (26-III). h = 1/16, 1/32, 1/64, 1/128from top to bottom respectively. λ = 1.

contrasted with other methods, e.g., that of [6], for which when λ < 1, regularizationwas needed in order to obtain convergence. Also, the results obtained by our methodbehave very similarly to those obtained in [12].

5.2. Generic examples with minimum L2(Σ)-norm boundary control. Inthe example discussed in Section 5.1, the minimum L2(Σ)-norm control is verysmooth. Using our methods, we obtained good approximations for this examplewithout the need for regularization. However, this is not the generic case. Ingeneral, even for smooth initial data, the minimum L2(Σ)-norm Dirichlet controlfor the controllability problem (1) will not be smooth. In this section, we illustratethis point and also examine the performance of our method for the generic case.

We choose Q = (0, 1)× (0, 1) in example I and Q = (0, 7/4)× (0, 1) in exampleII, III, and consider three sets of C∞(Ω) initial data:

(26)I. u0(x) = x(x− 1) and u1(x) = 0II. u0(x) = sin(πx) and u1(x) = π sin(πx)III. u0(x) = e

x and u1(x) = xex.


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f(u) = u3/2 and initial data (26-III). h = 1/16, 1/32, 1/64, 1/128from top to bottom respectively. λ = 1.

Note that the initial conditions (I), (II) vanish at the boundary and, that due tosymmetry, we have that u(t, 0) = u(t, 1), i.e., the control at the two sides of Q arethe same. For the initial conditions (III), we have that u(t, 0) ̸= u(t, 1).

First We examine the case λ = 1. In Figure 6, 8 and 10, we show the resultsfor the control for several grid sizes ranging from h = 1/16 to h = 1/1024. The(pointwise) convergence of the approximations is evident. Note that for the initialconditions given in (26), the minimum L2(Σ)-norm controls are seemingly piecewisesmooth, i.e., they contain jump discontinuities. The pointwise convergence of theapproximate control for the case of λ = 1 is probably a one-dimensional artifact; itis likely due to the fact that both the space and time variables in the wave equationin one dimension can act as time-like variables.

Further details about the computational results for the examples with initialconditions (I) given in (26) with λ = 4/5 are given in Table 2 and Figure 7. Theconvergence in L2(Σ) of the approximate minimum L2(Σ)-norm controls gh is ev-ident as is the convergence in L2(Q) of the approximate solution uh; the rates ofconvergence are seemingly first order.

Computational experiments were also carried out for λ = 7/8 for several valuesof the grid size ranging from h = 1/16 to h = 1/1024. The results are summarized


0 0.5 1 1.5 2 2.5 3−0.5

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Figure 14. gL, gR, uh(x, T ), and uht (x, T ) from left to right

with f(u) = ln(u2 + 1) and initial data (26-III). h =1/16, 1/32, 1/64, 1/128 from top to bottom respectively. λ = 1.

in Table 3 and 4. In Figures 9 and 11, we respectively provide, for the two sets ofinitial conditions (II) and (III), plots of the computed discrete solution gh, uh, uhtfor the two different values of λ and for different values of the grid size.

From Figures 9 and 11, we see that the approximate minimum L2(Σ)-normDirichlet controls obtained with values of λ < 1 are highly oscillatory. In fact, thefrequencies of the oscillations increase with decreasing grid size. However, it seemsthat the amplitudes of the oscillations do not increase as the grid size decreases.Furthermore, from the results in Table 3 and 4, it seems that for λ < 1, theapproximate controls gh, although oscillatory in nature and nonconvergent in apointwise sense, converge in an L2(Σ) sense.

The results of Table 2, 3, 4 and Figures 6, 7, 8, 9, 10, 11 indicate that forthe generic case of non-smooth minimum L2(Σ) controls and for general λ < 1,our method produces convergent (in L2(Q) and L2(Σ)) approximations without theneed of regulatization but the approximations are not in general convergent in apointwise sense. Of course, approximations that do not converge in a pointwisesense may be of little practical use, even if they converge in a root mean squaresense.


6. Computational experiments for controllability of semilinear wave equa-tions

We will again apply Algorithm 1 to the special case of V = 0, W = 0, Z = 0,α = 0 and σ, τ >> 1. We will test our algorithm with generic examples. If nonlinearterm f satisfies a certain property such as asymptotically linear or superlinear, thenthe exact control problem of the system 1 can be solvable;see, e.g., [4, 23, 24]. Inthis section, we examine the performance of our method for the asymptoticallylinear and superlinear cases.

We choose Q = (0, 3) × (0, 1) in example I, II, III, and consider three sets ofnonlinear term f :

(27)I. f(u) = sinuII. f(u) = u3/2

III. f(u) = ln(u2 + 1).

Note that we choose T = 3 for existence of control; see, e.g., [23, 24]. (I) is anexample of the asymptotically linear case and (II) is one of the superlinear case.(III) can be considered as either case. In general, we can not expect gL = gR dueto the nonlinear terms. We test the case λ = 1. The numerical approximationsby Algorithm 1 is convergent in L2 sense, that is, they have jump discontinuitiesas well. We will illustrate those through the figures 12, 13, and 14. For the linearcases, the number of iterations of the shooting methods is about 2 or 3, accordingto the tolerance and the accuracy of the machines we used. However the nonlinearcases are different and we need more iterations than the linear cases. We denotethe number of iterations as count. It is contained in the tables 5, 6 and 7 withL2(Σ)-norm of controls gh.

References

[1] R. Adams, Sobolev Spaces, Academic, New York, 1975.

[2] U. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical Solution of Boundary Value Prob-lems for Ordinary Differential Equations (1988), Prentice-Hall.

[3] W.H. Enright and K. Chow, Distributed parallel shooting for BVODEs, Proceedings ofHigh Performance Computing 1998, A. Tentner, ed., The Society for Computer Simulation,

Boston, 1998, pp. 203-210.[4] O. Yu. Emanuilov, Boundary controllability of semilinear evolution equations, Russian Math.

Surveys., 44 (1989) 183-184.[5] R. Glowinski and C.-H. Li, On the numerical implementation of the Hilbert Uniqueness

Method for the exact boundary controllability of the wave equation, C. R. Adad. Sc., Paris,T. 311, Serie I, 1990, pp. 135-142.

[6] R. Glowinski, C.-H. Li, and J.-L. Lions, A numerical approach to the exact boundary con-trollability of the wave equation (I). Dirichlet controls: description of the numerical methods,

Japan J. Appl. Math. 7 (1990) 1-76.[7] M. Gunzburger, L.S. Hou and L.-L. Ju, A numerical method for exact boundary controlla-

bility problems for the wave equation, submitted to Control, Optimization and Calculus ofVariations, 2002.

[8] M. Gunzburger and R. Nicolaides, An algorithm for the boundary control of the wave equa-tion, Appl. Math. Lett. 2 (1989) 225-228.

[9] J. A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the

1-D wave equation, Mathematical Modelling abd Numerical Analysis, 33 (1999) 407-438.[10] K. Jackson, A Survey of Parallel Numerical Methods for Initial Value Problems for Ordinary

Differential Equations, IEEE Transactions on Magnetics, 27 (1991) 3792-3797.[11] K. Jackson and S. Norsett, The potential for parallelism in Runge-Kutta methods. Part 1:

RK formulas in standard form, SIAM J. Numer. Anal., 32 (1995) 49-82.[12] L.-L. Ju, M. Gunzburger and L.S. Hou, Approximation of exact boundary controllability

problems for the 1-D wave equation by optimization-based methods, Contemp. Math., 330(2003) 133–153.


[13] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971.

[14] J.-L. Lions, Control of Distributed Singular Systems, Bordas, Paris, 1985.[15] J.-L. Lions, Exact controllability and stabilization and perturbations for distributed systems,

SIAM Review, 30 (1988) 1-68.[16] J.-L. Lions, Controlabilité Exacte, Perturbation et Stabilisation des Systèmes Destribués,

Vol. 1 and 2, Masson, Paris, 1988.

[17] M. Negreanu and E. Zuazua, Uniform boundary controllability of a discrete 1-D wave equa-tion, preprint, 2002.

[18] D. Russell, Controllability and stabilizability theory for linear partial differential equations:recent progress and open questions, SIAM Review 20 (1978) 639-739.

[19] D. Russell, A unified boundary controllability theory, Studies in Appl. Math. 52 (1973)189-211.

[20] D. Russell, Mathematics of Finite-Dimensional Control Systems – Theory and Design, Deck-ker, New York, 1979.

[21] J. W. Thomas, Numerical Partial Differential Equations – Finite Difference Methods,Springer, New York, 1998.

[22] E. Zuazua, Boundary observability for the finite difference space semi-discretizations of the2-D wave equation in the square, J. Math. Pures Appl., 78 (1999) 523-563.

[23] E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann.Inst. H. Poincar Anal. Non Linaire., 10 (1993) 109-129.

[24] E. Zuazua, Exact controllability for the semilinear wave equation, J. Math. Pures Appl., 69(1990) 1-31.

Department of Mathematics, Iowa State University, Ames, IA 50011, USA

E-mail : [email protected]: http://www.orion.math.iastate.edu/hou

Beijing Computational Science Research Center, No. 3, He-Qing Road, Hai-Dian District,Beijing, 100084, CHINA

E-mail : [email protected]: http://www.csrc.ac.cn/ jming/

National Institute of Meteological Research, 33 Seohobuk-ro, Seogwipo-si, Jeju-do, 697-845,Korea

E-mail : [email protected]

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